SECTION 1.1Sets of Real Numbers and

The Cartesian Coordinate Plane

SETS

A set is a well-defined collection of objects which are called the “elements” of the set

“Well-defined” means that it is possible to determine of something belongs to the collection or not

Example: collection of letters in the word “algebra”

Ways to describe sets: 1. The verbal method2. The roster method S = {a, l , g, e, b, r}3. The set-builder method S = { x | x is a letter in the world “algebra”}ScSa ,

SETS OF NUMBERS

1. The empty set Ø = { }2. The natural numbers N = {1, 2, 3, …}3. The whole numbers W = {0, 1, 2, … }4. The integers Z = { …, -2, -1, 0, 1, 2, …}5. The rational numbers Q =

Possesses a repeating or terminating decimal representation

6. The real numbers R = { x | x has a decimal representation }1. The irrational numbers P2. The complex numbers C

ZbandZab

a,

INTERVAL NOTATION

EXAMPLE

Set of Real Numbers

Interval Notation

Region on the Real Number Line

{ x | 1 ≤ x < 3 } [ 1 , 3 ) 1 3

{ x | -1 ≤ x ≤ 4 }

{ x | x ≤ 5 }

{ x | x > -2 }

INTERSECTION AND UNION

Suppose A and B are two sets The intersection of A and B:

The union of A and B

}|{ BxandAxxBA

}|{ BxorAxxBA

EXAMPLE

Express the following sets of numbers using interval notation

CARTESIAN COORDINATE PLANE

Two real number lines crossing at a right angle at 0

The horizontal number line is usually called the x-axis

The vertical number line is usually called the y-axis

CARTESIAN COORDINATES OF POINT

Imagine dropping a vertical line from the x-axis to P and extending a horizontal line from the y-axis to P

We describe the point P using the ordered pair (2,-4) The first number is called the abscissa or x-coordinate The second is called the ordinate or y-coordinate

The ordered pair (2,-4) comprise the Cartesian coordinates of the point P

EXAMPLE

Plot the following points: A(5,8) B(5/2, 3) C(-5.8, -3) D(4.5, -1) E(5,0) F(0,5) G(-7,0) H(0, -9) O(0,0)

IMPORTANT FACTS ABOUT THE CARTESIAN COORDINATE PLANE

(a,b) and (c,d) represent the same point in the plane if and only if a = c and b = d

(x,y) lies on the x-axis if and only if y = 0

(x,y) lies on the y-axis if and only if x = 0

The origin is the point (0,0). It is the only point common to both axes.

FOUR QUADRANTS

SYMMETRY

Two points (a,b) and (c,d) in the plane are said to be

Symmetric about the x-axis if a = c and b = -d

Symmetric about the y-axis if a = -c and b = d

Symmetric about the origin if a = -c and b = -d

SYMMETRY

P and S are symmetric about the x-axis, as are Q and R

P and Q are symmetric about the y-axis, as are R and S

P and R are symmetric about the origin, as are Q and S

EXAMPLE

Let P be the point (-2,3) Find the points which are symmetric to P about

the: x-axis y-axis origin

REFLECTIONS

To reflect a point (x,y) about the: x-axis replace y with –y y-axis replace x with –x origin replace x with -x and y with -y

DISTANCE IN PLANE

212

212

2 )()( yyxxd

212

212 )()( yyxxd

EXAMPLES

Find and simplify the distance between P(-2,3) and Q(1,-3)

Find all of the points with x-coordinate 1 which are 4 units from the point (3,2)

MIDPOINT FORMULA

2

,2

2121 yyxxM

EXAMPLE

Find the midpoint of the line segment connecting P(-2,3) and Q(1,-3)